Abstract:
In many fields, from theoretical mechanics to number theory, various canonical forms of transformations play a significant role. It follows from the theory of Jordan normal form that any one-parameter group of affine transformations of $n$-dimensional coordinate space is reduced by an affine transformation of coordinates to a triangular form. Is this true, if instead of affine transformations we consider arbitrary polynomial ones? In 1968, the affirmative answer for $n = 2$ was obtained by R. Rentschler. The negative answer for $n = 3$ was obtained in 1984 by H. Bass, and, for any $n > 2$, in 1986 by the speaker. One-parameter group is a special case of connected unipotent group, and polynomial transformation is a special case of rational one. For these more general groups and more general transformations, the corresponding general problem of the canonical form was formulated in 1984 by H. Bass. Although the groups of all rational transformations (called the Cremona groups) for $n > 1$ are infinite-dimensional, today it is clear that a number of basic concepts and properties of algebraic matrix groups are carried over to the Cremona groups. One of them is the notion of a Borel subgroup. H. Bass’s problem turns out to be naturally related to exploration of Borel subgroups of the Cremona groups. The talk is devoted to this range of topics.